Average Error: 52.6 → 0.2
Time: 12.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0580774043529224:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9441797120105823:\\ \;\;\;\;\left(x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0580774043529224:\\
\;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.9441797120105823:\\
\;\;\;\;\left(x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\

\end{array}
double f(double x) {
        double r3194896 = x;
        double r3194897 = r3194896 * r3194896;
        double r3194898 = 1.0;
        double r3194899 = r3194897 + r3194898;
        double r3194900 = sqrt(r3194899);
        double r3194901 = r3194896 + r3194900;
        double r3194902 = log(r3194901);
        return r3194902;
}
↓
double f(double x) {
        double r3194903 = x;
        double r3194904 = -1.0580774043529224;
        bool r3194905 = r3194903 <= r3194904;
        double r3194906 = -0.0625;
        double r3194907 = 5.0;
        double r3194908 = pow(r3194903, r3194907);
        double r3194909 = r3194906 / r3194908;
        double r3194910 = 0.125;
        double r3194911 = r3194903 * r3194903;
        double r3194912 = r3194903 * r3194911;
        double r3194913 = r3194910 / r3194912;
        double r3194914 = -0.5;
        double r3194915 = r3194914 / r3194903;
        double r3194916 = r3194913 + r3194915;
        double r3194917 = r3194909 + r3194916;
        double r3194918 = log(r3194917);
        double r3194919 = 0.9441797120105823;
        bool r3194920 = r3194903 <= r3194919;
        double r3194921 = -0.16666666666666666;
        double r3194922 = r3194912 * r3194921;
        double r3194923 = r3194903 + r3194922;
        double r3194924 = 0.075;
        double r3194925 = r3194908 * r3194924;
        double r3194926 = r3194923 + r3194925;
        double r3194927 = 0.5;
        double r3194928 = r3194927 / r3194903;
        double r3194929 = r3194928 - r3194913;
        double r3194930 = r3194903 + r3194929;
        double r3194931 = r3194930 + r3194903;
        double r3194932 = log(r3194931);
        double r3194933 = r3194920 ? r3194926 : r3194932;
        double r3194934 = r3194905 ? r3194918 : r3194933;
        return r3194934;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.6
Target45.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0580774043529224

    1. Initial program 61.8

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Taylor expanded around -inf 0.1

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{-1}{2}}{x} + \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right)}\]

    if -1.0580774043529224 < x < 0.9441797120105823

    1. Initial program 58.7

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{\left(x + \frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \frac{3}{40} \cdot {x}^{5}}\]

    if 0.9441797120105823 < x

    1. Initial program 30.7

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Taylor expanded around inf 0.2

      \[\leadsto \log \left(x + \color{blue}{\left(\left(x + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)}\right)\]

    3. Simplified0.2

      \[\leadsto \log \left(x + \color{blue}{\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0580774043529224:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} + \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} + \frac{\frac{-1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9441797120105823:\\ \;\;\;\;\left(x + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right) + {x}^{5} \cdot \frac{3}{40}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))